Pasha Zusmanovich - teaching

A.T. Fomenko, Double cover of the Klein bottle by torus

Pasha Zusmanovich

6/7TOPO Topology, winter semester 2025/2026

Monday 10:50-12:25 G302 ("Lectures" only)

LITERATURE
Main:

J.R. Munkres, Topology, 2nd ed., Prentice Hall, 2000
O.Ya. Viro, O.A. Ivanov, N.Yu. Netsvetaev, V.M. Kharlamov, Elementary Topology. Problem Textbook, AMS, 2008 Referred in what follows as [V]

Additional:

A.V. Arkhangelskii, V.I. Ponomarev, Fundamentals of General Topology: Problems and Exercises, D. Reidel, 1984
R. Engelking, General Topology, revised and completed ed., Heldermann Verlag, 1989
A. Fomenko, D. Fuchs, Homotopical Topology, 2nd ed., Springer, 2016 (included here solely for its pictures)
M.W. Hirsch, Differential Topology, Springer, 2001
J.F. Kelley, General Topology, Van Nostrand, 1955; reprinted by Springer, 1975 and Dover, 2017
C. Kosniowski, A First Course in Algebraic Topology, Cambridge Univ. Press, 1980 (Russian edition only)
J.W. Milnor, Topology from the Differentiable Viewpoint, The University Press of Virginia, 1965
S.A. Morris, Topology Without Tears
M. Starbird, F. Su, Topology Through Inquiry, MAA Press, 2019

All the books are available in electronic form in multiple places.

= available in the university library

A BRIEF SYNOPSIS
(Each class lasted approximately 1.5 hours; tutorials are running separately)

Class 1 September 22, 2025
Organizational details. The subject of topology. Definition of a topological space. Examples of topological spaces: trivial and discrete toplogy, topologies on 2- and 3-element sets, the standard topology on \(\mathbb R\). Number of topologies on a finite set.
([V], pp. xi-xii, 2, 11-12; Munkres, p.76).

Class 2 September 29, 2025
Base of a topology, examples. Any base of the standard topology on \(\mathbb R\) can be decreased. Criterion for a family of open sets to be a base ([V], p.16, 3.A).

Class 3 October 6, 2025
Further criteria for a family of subsets to be a base ([V], 3.B, 3.C). Finer and coarser topologies. Criterion for a topology to be coarser (Munkres, Lemma 13.3, also Theorem 1 here:

).
([V], pp.16-17; Munkres, p.81).

Class 4 October 13, 2025
The standard topology on \(\mathbb R^2\). Closed sets, definition of topology in terms of closed sets. Clopen sets. A parody of an episode from the movie "The Bunker" featuring clopen sets.
([V], pp.13-14,16-17).

A brief synopsis, videos, and homeworks from this course at the previous semesters

Created: Fri Oct 2 2020
Last modified: Mon Oct 13 2025 20:33:19 CEST